LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION

Yali Duan; Linghua Kong; Xianjin Chen; Min Guo

doi:10.11948/2018.1645

2018 Volume 8 Issue 6

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Yali Duan, Linghua Kong, Xianjin Chen, Min Guo. LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1645-1663. doi: 10.11948/2018.1645

Citation:

Yali Duan, Linghua Kong, Xianjin Chen, Min Guo. LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1645-1663. doi: 10.11948/2018.1645

LATTICE BOLTZMANN MODEL FOR TWO-DIMENSIONAL GENERALIZED SINE-GORDON EQUATION

1 School of Mathematical sciences, University of Science and Technology of China, Hefei, Anhui 230026, China;
2 School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China;
3 Department of Basic, North China College of Mechanics and Electrics, Changzhi, Shanxi, 046000, China

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Abstract

Abstract

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity u_t and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.
- Lattice Boltzmann method /
- Sine-Gordon equation /
- ChapmanEnskog expansion /
- soliton
MSC: 65M99;65Z05;74J30;74S30

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References

[1]	J. Argyris, M. Haase, J. C. Heinrich, Finite element approximation to twodimensional sine-Gordon solitons, Comput. Methods Appl. Mech. Eng., 1991, 86, 1-26. Google Scholar
[2]	R. Benzi, S. Succi, M. Vergassola, The lattice Boltzmann equation:theory and applications, Phys. Report, 1992, 222(3), 145-197. Google Scholar
[3]	A. G. Bratsos, An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions, Appl Numer Anal Comput Math, 2005, 2(2), 189-211. Google Scholar
[4]	A. G. Bratsos, A modified predictor-corrector scheme for the two-dimensional sine-Gordon equation, Numer Algorithms, 2006, 43, 295-308. Google Scholar
[5]	A. G. Bratsos, The solution of the two-dimensional sine-Gordon equation using the method of lines, J. Comput. Appl. Math., 2007, 206, 251-277. Google Scholar
[6]	A. G. Bratsos, A third order numerical scheme for the two-dimensional sineGordon equation, Math. Comput. Simulat., 2007, 76, 271-278. Google Scholar
[7]	P. Bhatnagar, E. Gross, M. Krook,A model for collision process in gas. I:Small amplitude processed in charged and neutral one component system, Phys. Rev., 1954, 94, 511-525. Google Scholar
[8]	S. Chen, G. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 1998, 30, 329-364. Google Scholar
[9]	P. L. Christiansen, P. S. Lomdahl, Numerical solution of 2+1 dimensional sine-Gordon solitons, Physica D, 1981, 2, 482-494. Google Scholar
[10]	J. G. Caputo, L. Loukitch, Dynamics of point Josephson junctions in a microstrip line, Physica C:Superconductivity, 2005, 425, 69-89. Google Scholar
[11]	T. P. Cheng, L. F. Li, Gauge Theory of Elementary Particle Physics, Claendon Press, Oxford, 2000. Google Scholar
[12]	P. L. Christiansen, O. H. Olsen, Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation, Phys Lett A, 1978, 68(2), 185-188. Google Scholar
[13]	M. Cui, High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation, Journal of Computational and Applied Mathematics, 2010, 235, 837-849. Google Scholar
[14]	S. P. Dawson, S. Chen, G. D. Doolen, Lattice Boltzmann computations for reaction-diffusion equations, J. Chem. Phys., 1993, 2, 1514-1523. Google Scholar
[15]	Y. Duan, R. Liu, Lattice Boltzmann model for two-dimensional unsteady Burgers' equation, J. Comput. Appl. Math., 2007, 206, 432-439. Google Scholar
[16]	Y. Duan, L. Kong, R. Zhang, A lattice Boitzmann model for the generalized Burgers-Hulexly equation, Physics A, 2012, 391, 625-632. Google Scholar
[17]	Y. Duan, X. Chen, L. Kong, Lattice Boltzmann model for the compound Burgers-Korteweg-de Vries equation, Chin. J. Comput. Phys., 2015, 32(6), 639-648. Google Scholar
[18]	Y. Duan, L. Kong, M. Guo, Numerical simulation of a class of nonlinear wave equations by lattice Boltzmann method, Commun. Math. Stat., 2017, 5, 13-35. Google Scholar
[19]	K. Djidjeli, W. G. Price, E. H. Twizell, Numerical solutions of a damped SineGordon equation in two space variables, J. Eng. Math., 1995, 29, 347-369. Google Scholar
[20]	Z. Dai, D. Xian, Homoclinic breather-wave solutions for sine-Gordon equation, Communications in Nonlinear Science and Numerical Simulation, 2009, 14, 3292-3295. Google Scholar
[21]	M. Dehghan, D. Mirzaei, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. Methods Appl. Mech. Eng., 2008, 197, 476-486. Google Scholar
[22]	J. A. Gonzalez, M. Martin-Landrove, Solitons in a nonlinear DNA model, Physics Letters A, 1994, 191, 409-415. Google Scholar
[23]	B. Guo, P. J. Pascual, M. J. Rodriguez, L. Vzquez, Numerical solution of the sine-Gordon equation, Appl. Math. Comput., 1986, 18, 1-14. Google Scholar
[24]	F. Higuera, S. Succi, R. Benzi, Lattice gas dynamics with enhanced collisions, Euro. Phys. Lett., 1989, 9, 345-349. Google Scholar
[25]	R. Hirota, Exact three-soliton solution of the two-dimensional sine-Gordon equation, J Phys Soc Jpn, 1973, 35, 1566. Google Scholar
[26]	J. D. Josephson, Supercurrents through barriers, Adv. Phys., 1965, 14, 419-451. Google Scholar
[27]	S. Johnson, P. Suarez, A. Biswas, New Exact Solutions for the Sine-Gordon Equation in 2+1 Dimensions, Computational Mathematics and Mathematical Physics, 2012, 52, 98-104. Google Scholar
[28]	P. Kaliappan, M. Lakshmanan Kadomtsev-Petviashvili, Two-dimensional sineGordon equations:reduction to Painlevé,transcendents, J Phys A:Math Gen, 1979, 12, 249-252. Google Scholar
[29]	G. Leibbrandt, New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions, Phys. Rev. Lett., 1978, 41, 435-438. Google Scholar
[30]	H. Lai, C. Ma, Lattice Boltzmann model for generalized nonlinear wave equation, Phys. Rev. E, 2011, 84, 046708. Google Scholar
[31]	L. Luo, The lattice-gas and lattice Boltzmann methods:past, present and future, Proceedings of International Conference on Applied Computational Fluid Dynamics. Beijing, China, October, Beijing, 2000, 52-83. Google Scholar
[32]	W. Liu, J. Sun, B. Wu, Space-time spectral method for the two-dimensional generalized sine-Gordon equation, J. Math.Anal.Appl., 2015, 427, 787-804. Google Scholar
[33]	D. Mirzaei, M. Dehghan, Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements, Engineering Analysis with Boundary Elements, 2009, 33, 12-24. Google Scholar
[34]	Y. Qian, S. Succi, S. Orszag, Recent advances in lattice Boltzmann computing, Annu. Rev. Comput. Phys., 1995, 3, 195-242. Google Scholar
[35]	J. Ram, P. Sapna, R. C. Mittal, Numerical simulation of two-dimensional sineGordon solitons by differential quadrature method, Computer Physics Communications, 2012, 183, 600-616. Google Scholar
[36]	B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusion equations, Phy. Rev. E, 2009, 79, 016701. Google Scholar
[37]	Q. Sheng, AQM Khaliq, D. A. Voss, Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. Comput. Simulation, 2005, 68, 355-373. Google Scholar
[38]	J. Xin, Modeling light bullets with the two-dimensional sine-Gordon equation, Physica D, 2000, 135, 345-368. Google Scholar
[39]	G. Yan, A lattice Boltzmann equation for waves, J. Comput. Phys., 2000, 161, 61-69. Google Scholar
[40]	J. Zhang, G. Yan, A lattice Boltzmann model for the Korteweg-de Vries equation with two conservation laws, Comput. Phys. Commun., 2009, 180, 1054-1062. Google Scholar
[41]	E. A. Zubova, N. K. Balabaev, Dynamics of soliton-like excitations in a chain of a polymer crystal:influence of neighbouring chains mobility, Journal of NonlinearMathematical Physics, 2001, 8, 305-311. Google Scholar